In general, an FSK transmitted signal S(t) may be mathematically expressed by the following equation: EQU S(t) = A cos (.omega..sub.c t + .phi.(t)) (1)
where A = signal amplitude
.phi.(t) = information-carrying phase function, and PA1 .omega..sub.c = carrier frequency.
In equation (1) above, the frequency of the transmitted sinusoid during any symbol-time is represented by the derivative of the argument (.omega..sub.c t + .phi.(t)). Thus, the instantaneous frequency .omega.(t) is expressed as: EQU .omega.(t) = .omega..sub.c + .phi.'(t), (2)
where .phi.'(t) is the frequency relative to the carrier frequency and is the information conveying frequency component. The .phi.(t) term of equation (1) is the phase of the transmitted sinusoid relative to the phase of a sinusoid of the carrier frequency .omega..sub.c. The actual phase function .phi.(t) resulting from the transmission of a particular sequence of information bits is defined as a transmitted "phase trajectory".
In FIG. 1, there are shown all possible phase trajectories which can result when any binary (M=2) FSK sequence of three bits is transmitted with a modulation index h, assuming that relative to the carrier .omega..sub.c the phase is zero at time t = 0. The particular path taken through the phase trajectory trellis shown in FIG. 1 is dependent upon the bit sequence transmitted. The binary bit sequence 1-0-1 is shown in FIG. 1 as following the phase trajectory path 0 - 2.pi.h, 2.pi.h-2.pi.h, 2.pi.h-4.pi.h, the phases at the ends of the bit times being prescribed phases shifted from each other by 2.pi.h radians. The phases at the ends of the bit times are referred to as "phase nodes". Thus, for each frequency Fi transmitted during a symbol time T.sub.s, there will be an associated prescribed phase shift between phase nodes at the respective beginning and end of the symbol time.
FIG. 2 illustrates an example of a transmitted bit represented by a frequency Fi. Two cycles of the frequency Fi are shown as making up the bit time with a zero phase difference between the beginning and end of the symbol time. For the three bit (101) example given below, there is a 2.pi.h radian phase shift for the first and third bits, and a zero phase shift for the second bit. Since, by definition, each bit is represented by a prescribed frequency (e.g. a "0" bit is represented as F0, and a "1" bit is represented as F1) by detecting the phase differences from the beginning to the end of each symbol time T.sub.s, a determination of the frequency transmitted during the symbol time can be effected. Thus, by detecting the phase differences for 0-2.pi.h, 2.pi.h-2.pi.h, 2.pi.h-4.pi.h, the transmitted signal F1-F0-F1 can be detected. Unfortunately, due to disturbances, such as those introduced by intersymbol interference, the phase nodes of the received FSK signals do not always coincide with their intended values of some multiple of 2.pi.h. Thus, the measured phases and, consequently, determined phase differences on the basis of such measured phases may lead to errors in frequency determination and, ultimately in detected FSK information.